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Theorem fin1a2lem6 8570
Description: Lemma for fin1a2 8580. Establish that  om can be broken into two equipollent pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypotheses
Ref Expression
fin1a2lem.b  |-  E  =  ( x  e.  om  |->  ( 2o  .o  x
) )
fin1a2lem.aa  |-  S  =  ( x  e.  On  |->  suc  x )
Assertion
Ref Expression
fin1a2lem6  |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E )

Proof of Theorem fin1a2lem6
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fin1a2lem.aa . . . 4  |-  S  =  ( x  e.  On  |->  suc  x )
21fin1a2lem2 8566 . . 3  |-  S : On
-1-1-> On
3 fin1a2lem.b . . . . 5  |-  E  =  ( x  e.  om  |->  ( 2o  .o  x
) )
43fin1a2lem4 8568 . . . 4  |-  E : om
-1-1-> om
5 f1f 5603 . . . 4  |-  ( E : om -1-1-> om  ->  E : om --> om )
6 frn 5562 . . . . 5  |-  ( E : om --> om  ->  ran 
E  C_  om )
7 omsson 6479 . . . . 5  |-  om  C_  On
86, 7syl6ss 3365 . . . 4  |-  ( E : om --> om  ->  ran 
E  C_  On )
94, 5, 8mp2b 10 . . 3  |-  ran  E  C_  On
10 f1ores 5652 . . 3  |-  ( ( S : On -1-1-> On  /\ 
ran  E  C_  On )  ->  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( S " ran  E
) )
112, 9, 10mp2an 667 . 2  |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( S " ran  E )
129sseli 3349 . . . . . . . . 9  |-  ( b  e.  ran  E  -> 
b  e.  On )
131fin1a2lem1 8565 . . . . . . . . 9  |-  ( b  e.  On  ->  ( S `  b )  =  suc  b )
1412, 13syl 16 . . . . . . . 8  |-  ( b  e.  ran  E  -> 
( S `  b
)  =  suc  b
)
1514eqeq1d 2449 . . . . . . 7  |-  ( b  e.  ran  E  -> 
( ( S `  b )  =  a  <->  suc  b  =  a
) )
1615rexbiia 2746 . . . . . 6  |-  ( E. b  e.  ran  E
( S `  b
)  =  a  <->  E. b  e.  ran  E  suc  b  =  a )
174, 5, 6mp2b 10 . . . . . . . . . . . 12  |-  ran  E  C_ 
om
1817sseli 3349 . . . . . . . . . . 11  |-  ( b  e.  ran  E  -> 
b  e.  om )
19 peano2 6495 . . . . . . . . . . 11  |-  ( b  e.  om  ->  suc  b  e.  om )
2018, 19syl 16 . . . . . . . . . 10  |-  ( b  e.  ran  E  ->  suc  b  e.  om )
213fin1a2lem5 8569 . . . . . . . . . . . 12  |-  ( b  e.  om  ->  (
b  e.  ran  E  <->  -. 
suc  b  e.  ran  E ) )
2221biimpd 207 . . . . . . . . . . 11  |-  ( b  e.  om  ->  (
b  e.  ran  E  ->  -.  suc  b  e. 
ran  E ) )
2318, 22mpcom 36 . . . . . . . . . 10  |-  ( b  e.  ran  E  ->  -.  suc  b  e.  ran  E )
2420, 23jca 529 . . . . . . . . 9  |-  ( b  e.  ran  E  -> 
( suc  b  e.  om 
/\  -.  suc  b  e. 
ran  E ) )
25 eleq1 2501 . . . . . . . . . 10  |-  ( suc  b  =  a  -> 
( suc  b  e.  om  <->  a  e.  om ) )
26 eleq1 2501 . . . . . . . . . . 11  |-  ( suc  b  =  a  -> 
( suc  b  e.  ran  E  <->  a  e.  ran  E ) )
2726notbid 294 . . . . . . . . . 10  |-  ( suc  b  =  a  -> 
( -.  suc  b  e.  ran  E  <->  -.  a  e.  ran  E ) )
2825, 27anbi12d 705 . . . . . . . . 9  |-  ( suc  b  =  a  -> 
( ( suc  b  e.  om  /\  -.  suc  b  e.  ran  E )  <-> 
( a  e.  om  /\ 
-.  a  e.  ran  E ) ) )
2924, 28syl5ibcom 220 . . . . . . . 8  |-  ( b  e.  ran  E  -> 
( suc  b  =  a  ->  ( a  e. 
om  /\  -.  a  e.  ran  E ) ) )
3029rexlimiv 2833 . . . . . . 7  |-  ( E. b  e.  ran  E  suc  b  =  a  ->  ( a  e.  om  /\ 
-.  a  e.  ran  E ) )
31 peano1 6494 . . . . . . . . . . . . . 14  |-  (/)  e.  om
323fin1a2lem3 8567 . . . . . . . . . . . . . 14  |-  ( (/)  e.  om  ->  ( E `  (/) )  =  ( 2o  .o  (/) ) )
3331, 32ax-mp 5 . . . . . . . . . . . . 13  |-  ( E `
 (/) )  =  ( 2o  .o  (/) )
34 om0x 6955 . . . . . . . . . . . . 13  |-  ( 2o 
.o  (/) )  =  (/)
3533, 34eqtri 2461 . . . . . . . . . . . 12  |-  ( E `
 (/) )  =  (/)
36 f1fun 5605 . . . . . . . . . . . . . 14  |-  ( E : om -1-1-> om  ->  Fun 
E )
374, 36ax-mp 5 . . . . . . . . . . . . 13  |-  Fun  E
38 f1dm 5607 . . . . . . . . . . . . . . 15  |-  ( E : om -1-1-> om  ->  dom 
E  =  om )
394, 38ax-mp 5 . . . . . . . . . . . . . 14  |-  dom  E  =  om
4031, 39eleqtrri 2514 . . . . . . . . . . . . 13  |-  (/)  e.  dom  E
41 fvelrn 5836 . . . . . . . . . . . . 13  |-  ( ( Fun  E  /\  (/)  e.  dom  E )  ->  ( E `  (/) )  e.  ran  E )
4237, 40, 41mp2an 667 . . . . . . . . . . . 12  |-  ( E `
 (/) )  e.  ran  E
4335, 42eqeltrri 2512 . . . . . . . . . . 11  |-  (/)  e.  ran  E
44 eleq1 2501 . . . . . . . . . . 11  |-  ( a  =  (/)  ->  ( a  e.  ran  E  <->  (/)  e.  ran  E ) )
4543, 44mpbiri 233 . . . . . . . . . 10  |-  ( a  =  (/)  ->  a  e. 
ran  E )
4645necon3bi 2650 . . . . . . . . 9  |-  ( -.  a  e.  ran  E  ->  a  =/=  (/) )
47 nnsuc 6492 . . . . . . . . 9  |-  ( ( a  e.  om  /\  a  =/=  (/) )  ->  E. b  e.  om  a  =  suc  b )
4846, 47sylan2 471 . . . . . . . 8  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  E. b  e.  om  a  =  suc  b )
49 eleq1 2501 . . . . . . . . . . . . . . . 16  |-  ( a  =  suc  b  -> 
( a  e.  om  <->  suc  b  e.  om )
)
50 eleq1 2501 . . . . . . . . . . . . . . . . 17  |-  ( a  =  suc  b  -> 
( a  e.  ran  E  <->  suc  b  e.  ran  E ) )
5150notbid 294 . . . . . . . . . . . . . . . 16  |-  ( a  =  suc  b  -> 
( -.  a  e. 
ran  E  <->  -.  suc  b  e. 
ran  E ) )
5249, 51anbi12d 705 . . . . . . . . . . . . . . 15  |-  ( a  =  suc  b  -> 
( ( a  e. 
om  /\  -.  a  e.  ran  E )  <->  ( suc  b  e.  om  /\  -.  suc  b  e.  ran  E ) ) )
5352anbi1d 699 . . . . . . . . . . . . . 14  |-  ( a  =  suc  b  -> 
( ( ( a  e.  om  /\  -.  a  e.  ran  E )  /\  b  e.  om ) 
<->  ( ( suc  b  e.  om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om ) ) )
54 simplr 749 . . . . . . . . . . . . . . 15  |-  ( ( ( suc  b  e. 
om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om )  ->  -.  suc  b  e. 
ran  E )
5521adantl 463 . . . . . . . . . . . . . . 15  |-  ( ( ( suc  b  e. 
om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om )  ->  ( b  e.  ran  E  <->  -.  suc  b  e.  ran  E ) )
5654, 55mpbird 232 . . . . . . . . . . . . . 14  |-  ( ( ( suc  b  e. 
om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om )  ->  b  e.  ran  E
)
5753, 56syl6bi 228 . . . . . . . . . . . . 13  |-  ( a  =  suc  b  -> 
( ( ( a  e.  om  /\  -.  a  e.  ran  E )  /\  b  e.  om )  ->  b  e.  ran  E ) )
5857com12 31 . . . . . . . . . . . 12  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  b  e. 
om )  ->  (
a  =  suc  b  ->  b  e.  ran  E
) )
5958impr 616 . . . . . . . . . . 11  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  b  e.  ran  E )
60 simprr 751 . . . . . . . . . . . 12  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  a  =  suc  b )
6160eqcomd 2446 . . . . . . . . . . 11  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  suc  b  =  a )
6259, 61jca 529 . . . . . . . . . 10  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  ( b  e. 
ran  E  /\  suc  b  =  a ) )
6362ex 434 . . . . . . . . 9  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  ( (
b  e.  om  /\  a  =  suc  b )  ->  ( b  e. 
ran  E  /\  suc  b  =  a ) ) )
6463reximdv2 2823 . . . . . . . 8  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  ( E. b  e.  om  a  =  suc  b  ->  E. b  e.  ran  E  suc  b  =  a ) )
6548, 64mpd 15 . . . . . . 7  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  E. b  e.  ran  E  suc  b  =  a )
6630, 65impbii 188 . . . . . 6  |-  ( E. b  e.  ran  E  suc  b  =  a  <->  ( a  e.  om  /\  -.  a  e.  ran  E ) )
6716, 66bitri 249 . . . . 5  |-  ( E. b  e.  ran  E
( S `  b
)  =  a  <->  ( a  e.  om  /\  -.  a  e.  ran  E ) )
68 f1fn 5604 . . . . . . 7  |-  ( S : On -1-1-> On  ->  S  Fn  On )
692, 68ax-mp 5 . . . . . 6  |-  S  Fn  On
70 fvelimab 5744 . . . . . 6  |-  ( ( S  Fn  On  /\  ran  E  C_  On )  ->  ( a  e.  ( S " ran  E
)  <->  E. b  e.  ran  E ( S `  b
)  =  a ) )
7169, 9, 70mp2an 667 . . . . 5  |-  ( a  e.  ( S " ran  E )  <->  E. b  e.  ran  E ( S `
 b )  =  a )
72 eldif 3335 . . . . 5  |-  ( a  e.  ( om  \  ran  E )  <->  ( a  e. 
om  /\  -.  a  e.  ran  E ) )
7367, 71, 723bitr4i 277 . . . 4  |-  ( a  e.  ( S " ran  E )  <->  a  e.  ( om  \  ran  E
) )
7473eqriv 2438 . . 3  |-  ( S
" ran  E )  =  ( om  \  ran  E )
75 f1oeq3 5631 . . 3  |-  ( ( S " ran  E
)  =  ( om 
\  ran  E )  ->  ( ( S  |`  ran  E ) : ran  E -1-1-onto-> ( S " ran  E
)  <->  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E
) ) )
7674, 75ax-mp 5 . 2  |-  ( ( S  |`  ran  E ) : ran  E -1-1-onto-> ( S
" ran  E )  <->  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om 
\  ran  E )
)
7711, 76mpbi 208 1  |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   E.wrex 2714    \ cdif 3322    C_ wss 3325   (/)c0 3634    e. cmpt 4347   Oncon0 4715   suc csuc 4717   dom cdm 4836   ran crn 4837    |` cres 4838   "cima 4839   Fun wfun 5409    Fn wfn 5410   -->wf 5411   -1-1->wf1 5412   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090   omcom 6475   2oc2o 6910    .o comu 6914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-omul 6921
This theorem is referenced by:  fin1a2lem7  8571
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